Nonlocal meta-lens with Huygens’ bound states in the continuum

Meta-lenses composed of artificial meta-atoms have stimulated substantial interest due to their compact and flexible wavefront shaping capabilities, outperforming bulk optical devices. The operating bandwidth is a critical factor determining the meta-lens’ performance across various wavelengths. Meta-lenses that operate in a narrowband manner relying on nonlocal effects can effectively reduce disturbance and crosstalk from non-resonant wavelengths, making them well-suitable for specialized applications such as nonlinear generation and augmented reality/virtual reality display. However, nonlocal meta-lenses require striking a balance between local phase manipulation and nonlocal resonance excitation, which involves trade-offs among factors like quality-factor, efficiency, manipulation dimensions, and footprint. In this work, we experimentally demonstrate the nonlocal meta-lens featuring Huygens’ bound states in the continuum (BICs) and its near-infrared imaging application. All-dielectric integrated-resonant unit is particularly optimized to efficiently induce both the quasi-BIC and generalized Kerker effect, while ensuring the rotation-angle robustness for generating geometric phase. The experimental results show that the single-layer nonlocal Huygens’ meta-lens possesses a high quality-factor of 104 and achieves a transmission polarization conversion efficiency of 55%, exceeding the theoretical limit of 25%. The wavelength-selective two-dimensional focusing and imaging are demonstrated as well. This work will pave the way for efficient nonlocal wavefront shaping and meta-devices.

Fig. 6a and b show the multipole decompositions of periodic IRU and single IRU, respectively.
Low-Q Mie resonance (MDxy) and high-Q q-BIC mode (MDz) can both be excited in the period IRU.For a single IRU, without the nonlocal effect provided by the coupling with adjacent units, only the local Mie resonance can be excited.The relationship between the Q-factor and the number of meta-units along the radial axis in the meta-lens is provided in Supplementary Fig. 6c.It is evident that the Q-factor rises with the increase in the number of meta-units along the radial direction, highlighting the presence of a nonlocal effect within the meta-lens, even when the building blocks possess different structural orientations.The Q-factor increases as the number of meta-units grows, and it stabilizes at ~120 when the unit count exceeds 50.This further confirms the presence of the nonlocal effect and provides guidance for determining the meta-lens size.

Supplementary Note 7: Discussion on the theoretical limit in nonlocal meta-lens
As discussed in the main context, the metasurface can be conceptualized as a four-port system.In such cases, the transmission coefficient of cross-polarization conversion tLR can be derived as 1,2   [ ] Here, tij represents the transmission coefficient for i-polarized light when the metasurface is illuminated with j-polarized light.The subscripts L and R signify left-handed polarization and right-handed polarization, respectively.Referring to Supplementary Equation ( 1), the maximum cross-polarization conversion efficiency TLR = |tLR| 2 reaches 25% when tLL = 50%.
In this work, we abstain from introducing guide-mode resonance within the substrate or utilizing meta-atoms with wavelength-level height.Consequently, our proposed approach does not hinge on breaking out-of-plane symmetry to surpass the 25% polarization conversion efficiency threshold.Furthermore, the predominance of vertical magnetic dipole (MD) moments in the quasibound state in the continuum (q-BIC) mode, primarily contributed by the in-plane current, lends support to our claim: the disruption of out-of-plane symmetry in our design is minimal.To further illustrate that our proposed approach for achieving >25% polarization conversion efficiency in an optically thin metasurface is attributable to the involvement of both the MD and q-BIC, we present the polarization conversion efficiencies TLR for various offsets L that control the spectral separation between MD and q-BIC.As can be seen in Fig. 2b in the main article and Supplementary Fig. 7, when the offset L ranges between 270 nm and 400 nm, the Mie-type MD resonance couples with the q-BIC mode, resulting in the generalized Kerker effect.Consequently, the transmission polarization conversion efficiency exceeds 25%, reaching a maximum of 85.4% with L = 310 nm.
However, when L exceeds 400 nm or falls below 270 nm, the two resonant wavelengths are significantly distant, and the single mode of q-BIC can only achieve an efficiency limited to approximately 25%.If the designed unit possesses an intrinsically large out-of-plane asymmetry, the efficiencies for all offsets should not be limited to 25%, and the rapid increase around L = 310 nm would not occur.These observations confirm the intrinsic weak out-of-plane asymmetry of the designed unit and underscore the importance of the Kerker effect.
into the pioneering aspects and advancements of our methodology on the pioneering design of IRU (also refer to Supplementary Fig. 8).
(a) Amplified Q-factor: By disrupting the structure's in-plane symmetry, we usher in symmetry-protected q-BICs, lending a significant boost to the Q-factor.This augmentation is achieved by fine-tuning the asymmetric parameter, affording flexibility in manipulation.(c) Enhanced transmission efficiency: Beyond the q-BIC mode, our innovation encompasses the incorporation of a Mie-type MD resonance.This synergistic coupling with q-BIC engenders the generalized Kerker effect, fostering unidirectional transmission and surpassing the theoretical limit of 25% for transmissive circular polarization conversion.
Concurrently attaining these three pivotal points presents a formidable challenge within the realm of conventional meta-units characterized by a single resonant mode and lackluster structure.
Moreover, the optimization of IRUs warrants a delicate equilibrium, wherein geometric shape We have compared our work with Ref. [5] in the main text because both two works are singlelayer nonlocal meta-lenses for two-dimensional wavefront shaping.To provide a more comprehensive comparison, a comparison with another similar work of Ref. [7] is also given as follows.It proposed a novel phase gradient metasurface made entirely from individually addressable high-Q-factor meta-atoms.Although the Q-factor of ~380 reported in Ref. [7] is higher than our work and the efficiency of 59% is comparable, it is important to highlight the advantages and distinctions of our work as follows: (a) Advantages i.
Our work achieves two-dimensional wavefront shaping with high Q-factor and efficiency, surpassing the previous work's demonstration of only one-dimensional control 7 .The (b) Stalwart rotation angle robustness: The rotation angle assumes paramount importance in generating the geometric phase essential for meta-lens functionality.Ensuring the steadfastness of resonance characteristics -Q-factor, efficiency, phase, and resonant wavelength -across varying rotation angles is imperative for seamless phase modulation and resonance excitation.